The Judiciary Committee's Subcommittee on Constitutional Amendments was under the chairmanship of Estes Kefauver when Bayh entered the Senate. When Kefauver died unexpectedly in August 1963, Judiciary Committee Chair James Eastland intended to dissolve the Subcommittee for budgetary reasons, but Bayh persuaded him to let it continue under Bayh's chairmanship if it could be done with minimal expense. Staff members describe the tiny corner of the Bayh suite in which it was originally housed, with Chief Counsel Larry Conrad's desk in a closet.

With the assassination of President Kennedy in November 1963, the subcommittee was thrust into the limelight as the inadequacies of present law to handle Presidential succession and vacancy in the vice presidency became evident. By 1965, Bayh and his staff had picked up on the multitude of issues Kefauver had been pursuing, and over the next 15 years not only managed passage and ratification of the 25th and 26th Amendments to the Constitution but tackled reform of the electoral college, the possibility of direct election of the President, reapportionment, and equal rights for women.

Following discussions begun in late 1976, the Judiciary Committee's subcommittees were reorganized at the end of March 1977, and the Subcommittees on Constitutional Amendments and on Constitutional Rights were combined to form the Subcommittee on the Constitution. In 1978, issues related to juvenile justice previously handled by the Subcommittee on Juvenile Delinquency, which Bayh had chaired since 1970, were incorporated into the jurisdiction of the Subcommittee on the Constitution as relating to children's rights while other matters relating to juvenile delinquency were transferred to the Criminal Laws Subcommittee. Files of the Subcommittee on Juvenile Delinquency are in a separate series under that name.